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A convergent and asymptotic Laplace method for integrals

dc.contributor.authorLópez García, José Luis
dc.contributor.authorPagola Martínez, Pedro Jesús
dc.contributor.authorPalacios Herrero, Pablo
dc.contributor.departmentEstatistika, Informatika eta Matematikaeu
dc.contributor.departmentInstitute for Advanced Materials and Mathematics - INAMAT2en
dc.contributor.departmentEstadística, Informática y Matemáticases_ES
dc.contributor.funderUniversidad Pública de Navarra / Nafarroako Unibertsitate Publikoaes
dc.date.accessioned2022-11-14T08:43:52Z
dc.date.available2022-11-14T08:43:52Z
dc.date.issued2023
dc.date.updated2022-11-14T07:57:32Z
dc.description.abstractWatson’s lemma and Laplace’s method provide asymptotic expansions of Laplace integrals F (z) := ∫ ∞ 0 e −zf (t) g(t)dt for large values of the parameter z. They are useful tools in the asymptotic approximation of special functions that have a Laplace integral representation. But in most of the important examples of special functions, the asymptotic expansion derived by means of Watson’s lemma or Laplace’s method is not convergent. A modification of Watson’s lemma was introduced in [Nielsen, 1906] where, by the use of inverse factorial series, a new asymptotic as well as convergent expansion of F (z), for the particular case f (t) = t, was derived. In this paper we go some steps further and investigate a modification of the Laplace’s method for F (z), with a general phase function f (t), to derive asymptotic expansions of F (z) that are also convergent, accompanied by error bounds. An analysis of the remainder of this new expansion shows that it is convergent under a mild condition for the functions f (t) and g(t), namely, these functions must be analytic in certain starlike complex regions that contain the positive axis [0,∞). In many practical situations (in many examples of special functions), the singularities of f (t) and g(t) are off this region and then this method provides asymptotic expansions that are also convergent. We illustrate this modification of the Laplace’s method with the parabolic cylinder function U(a, z), providing an asymptotic expansions of this function for large z that is also convergent.en
dc.description.sponsorshipThis research was supported by the Universidad Pública de Navarra, grant PRO-UPNA (6158) 01/01/2022. Open access funding provided by Universidad Pública de Navarra.en
dc.format.mimetypeapplication/pdfen
dc.identifier.citationLópez, J. L., Pagola, P. J., & Palacios, P. (2023). A convergent and asymptotic Laplace method for integrals. Journal of Computational and Applied Mathematics, 422, 114897.en
dc.identifier.doi10.1016/j.cam.2022.114897
dc.identifier.issn0377-0427
dc.identifier.urihttps://academica-e.unavarra.es/handle/2454/44322
dc.language.isoengen
dc.publisherElsevieren
dc.relation.ispartofJournal of Computational and Applied Mathematics 422 (2023) 114897en
dc.relation.publisherversionhttps://doi.org/10.1016/j.cam.2022.114897
dc.rights© 2022 The Author(s). This is an open access article under the CC BY-NC-ND licenseen
dc.rights.accessRightsinfo:eu-repo/semantics/openAccess
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/
dc.subjectAsymptotic expansions of integralsen
dc.subjectWatson’s lemmaen
dc.subjectLaplace’s methoden
dc.subjectConvergent expansionsen
dc.subjectSpecial functionsen
dc.titleA convergent and asymptotic Laplace method for integralsen
dc.typeinfo:eu-repo/semantics/article
dc.type.versionVersión publicada / Argitaratu den bertsioaes
dc.type.versioninfo:eu-repo/semantics/publishedVersionen
dspace.entity.typePublication
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relation.isAuthorOfPublication68ff8840-f80e-4119-ac1a-edfad578de07
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relation.isAuthorOfPublication.latestForDiscoverye6cd33c5-6d5e-455c-b8da-32a9702e16c8

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